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Relevant bibliographies by topics / Gauss-Lobatto

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Academic literature on the topic 'Gauss-Lobatto'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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Journal articles on the topic "Gauss-Lobatto"

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Bhrawy, A. H., M. A. Alghamdi, and D. Baleanu. "Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/513808.

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The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-legθ-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

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Gautschi, Walter. "Generalized Gauss?Radau and Gauss?Lobatto Formulae." BIT Numerical Mathematics 44, no. 4 (2004): 711–20. http://dx.doi.org/10.1007/s10543-004-3812-0.

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Pejcev, Aleksandar, and Ljubica Mihic. "Errors of Gauss-Radau and Gauss-Lobatto quadratures with double end point." Applicable Analysis and Discrete Mathematics 13, no. 2 (2019): 463–77. http://dx.doi.org/10.2298/aadm180408011p.

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Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315{329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss- Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors.

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Wan, Zhengsu, Benyu Guo, and Chengjian Zhang. "Generalized Jacobi-Gauss-Lobatto interpolation." Frontiers of Mathematics in China 8, no. 4 (2013): 933–60. http://dx.doi.org/10.1007/s11464-013-0271-4.

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Calvetti, Daniela, and Lothar Reichel. "Symmetric Gauss–Lobatto and Modified Anti-Gauss Rules." BIT Numerical Mathematics 43, no. 3 (2003): 541–54. http://dx.doi.org/10.1023/b:bitn.0000007053.03860.c0.

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Coluccio, Loredana, Alfredo Eisinberg, and Giuseppe Fedele. "Gauss–Lobatto to Bernstein polynomials transformation." Journal of Computational and Applied Mathematics 222, no. 2 (2008): 690–700. http://dx.doi.org/10.1016/j.cam.2007.12.007.

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Welfert, B. D. "A note on classical Gauss–Radau and Gauss–Lobatto quadratures." Applied Numerical Mathematics 60, no. 6 (2010): 637–44. http://dx.doi.org/10.1016/j.apnum.2010.03.006.

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Li, Shikang. "Kronrod Extension of Generalized Gauss-Radau and Gauss-Lobatto Formulae." Rocky Mountain Journal of Mathematics 26, no. 4 (1996): 1455–72. http://dx.doi.org/10.1216/rmjm/1181071998.

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Gautschi, Walter, and Shikang Li. "Gauss—Radau and Gauss—Lobatto quadratures with double end points." Journal of Computational and Applied Mathematics 34, no. 3 (1991): 343–60. http://dx.doi.org/10.1016/0377-0427(91)90094-z.

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Mihic, Ljubica, Aleksandar Pejcev, and Miodrag Spalevic. "Error bounds for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind." Filomat 30, no. 1 (2016): 231–39. http://dx.doi.org/10.2298/fil1601231m.

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For analytic functions the remainder terms of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points -+1, for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi and Li in paper [The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, Journal of Computational and Applied Mathematics 33 (1990) 315-329.]

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Dissertations / Theses on the topic "Gauss-Lobatto"

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Garcia, Maxine Patricia. "Collocation methods for mixed order boundary value problems." Thesis, Imperial College London, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.322404.

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Small, Scott Joseph. "Runge-Kutta type methods for differential-algebraic equations in mechanics." Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1082.

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Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3. Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods. When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s. Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators. This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.

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Moraru, Laurentiu Eugen. "Numerical Predictions and Measurements in the Lubrication of Aeronautical Engine and Transmission Components." University of Toledo / OhioLINK, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1125769629.

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Derflinger, Gerhard, Wolfgang Hörmann, and Josef Leydold. "Random Variate Generation by Numerical Inversion when only the Density Is Known." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2008. http://epub.wu.ac.at/1112/1/document.pdf.

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We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. (author´s abstract)<br>Series: Research Report Series / Department of Statistics and Mathematics

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Derflinger, Gerhard, Wolfgang Hörmann, and Josef Leydold. "Online Supplement to "Random Variate Generation by Numerical Inversion When Only the Density Is Known"." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2009. http://epub.wu.ac.at/162/1/document.pdf.

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We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and nearly the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. <P> This paper is the revised final version of the working paper no. 78 of this research report series.<br>Series: Research Report Series / Department of Statistics and Mathematics

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Book chapters on the topic "Gauss-Lobatto"

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Shang, J. S., and P. G. Huang. "A Local Resolution Refinement Algorithm Using Gauss–Lobatto Quadrature." In Computational Fluid Dynamics 2010. Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-17884-9_65.

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Argoubi, Mohamed Ali, Mohamed Trabelssi, and Molka Chiboub Hili. "An Adapted Formulation for the Locally Adaptive Weak Quadrature Element Method Using Gauss-Lobatto Points." In Applied Condition Monitoring. Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-34190-8_33.

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Parikh, Amit K., and Jishan K. Shaikh. "Numerical Solution of Counter-Current Imbibition Phenomenon in Homogeneous Porous Media Using Polynomial Base Differential Quadrature Method with Chebyshev-Gauss-Lobatto Grid Points." In Advances in Intelligent Systems and Computing. Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-15-9953-8_17.

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Chan, Raymond H., Chen Greif, and Dianne P. O’Leary. "Some Modified Matrix Eigenvalue Problems." In Milestones In Matrix Computation: Selected Works Of Gene H. Golub, With Commentaries. Oxford University PressOxford, 2007. http://dx.doi.org/10.1093/oso/9780199206810.003.0027.

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Abstract We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss-Radau and Gauss-Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.

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Gautschi, Walter. "Applications." In Orthogonal Polynomials. Oxford University Press, 2004. http://dx.doi.org/10.1093/oso/9780198506720.003.0005.

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The connection between orthogonal polynomials and quadrature rules has already been elucidated in §1.4. The centerpieces were the Gaussian quadrature rule and its close relatives—the Gauss–Radau and Gauss–Lobatto rules (§1.4.2). There are, however, a number of further extensions of Gauss’s approach to numerical quadrature. Principal among them are Kronrod’s idea of extending an n-point Gauss rule to a (2n + 1)-point rule by inserting n + 1 additional nodes and choosing all weights in such a way as to maximize the degree of exactness (cf. Definition 1.44), and Turán’s extension of the Gauss quadrature rule allowing not only function values, but also derivative values, to appear in the quadrature sum. More recent extensions relate to the concept of accuracy, requiring exactness not only for polynomials of a certain degree, but also for rational functions with prescribed poles. Gauss quadrature can also be adapted to deal with Cauchy principal value integrals, and there are other applications of Gauss’s ideas, for example, in combination with Stieltjes’s procedure or the modified Chebyshev algorithm, to generate polynomials orthogonal on several intervals, or, in comnbination with Lanczos’s algorithm, to estimate matrix functionals. The present section is to discuss these questions in turn, with computational aspects foremost in our mind. We have previously seen in Chapter 2 how Gauss quadrature rules can be effectively employed in the context of computational methods; for example, in computing the absolute and relative condition numbers of moment maps (§2.1.5), or as a means of discretizing measures in the multiple-component discretization method for orthogonal polynomials (§2.2.5) and in Stieltjes-type methods for Sobolev orthogonal polynomials (§2.5.2). It is time now to discuss the actual computation of these, and related, quadrature rules.

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El–Baghdady, Galal I., M. S. El–Azab, and W. S. El–Beshbeshy. "Determination of Legendre–Gauss–Lobatto Pseudo–spectral Method for One–Dimensional Advection–Diffusion Equation." In Current Topics on Mathematics and Computer Science Vol. 4. Book Publisher International (a part of SCIENCEDOMAIN International), 2021. http://dx.doi.org/10.9734/bpi/ctmcs/v4/10245d.

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Conference papers on the topic "Gauss-Lobatto"

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Zhao, Zhenyu, and Lei You. "Numerical Differentiation by Legendre-Gauss-Lobatto Interpolation." In 2010 International Conference on Computational and Information Sciences (ICCIS). IEEE, 2010. http://dx.doi.org/10.1109/iccis.2010.196.

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Trimbitas, Radu Tiberiu, and Maria Gabriela Trimbitas. "Gauss-Lobatto-Kronrod Formulae and Adaptive Numerical Integration." In 2008 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. IEEE, 2008. http://dx.doi.org/10.1109/synasc.2008.27.

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Ortleb, Sigrun. "A Fourier-type analysis of the Gauss and Gauss-Lobatto P1-discontinuous Galerkin methods for the linear advection-diffusion equation." In CENTRAL EUROPEAN SYMPOSIUM ON THERMOPHYSICS 2019 (CEST). AIP Publishing, 2019. http://dx.doi.org/10.1063/1.5114349.

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Al-Humedi, Hameeda Oda, and Firas Amer Al-Saadawi. "Spectral shifted Jacobi-Gauss-Lobatto methodology for solving two-dimensional time-space fractional bioheat model." In INTERNATIONAL CONFERENCE ON EMERGING APPLICATIONS IN MATERIAL SCIENCE AND TECHNOLOGY: ICEAMST 2020. AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0007638.

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Ying, Teh Yuan, and Nazeeruddin Yaacob. "Numerical solution of first order initial value problem using 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method." In PROCEEDINGS OF THE 20TH NATIONAL SYMPOSIUM ON MATHEMATICAL SCIENCES: Research in Mathematical Sciences: A Catalyst for Creativity and Innovation. AIP, 2013. http://dx.doi.org/10.1063/1.4801122.

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Nelson, Daniel A., and Gustaaf B. Jacobs. "Computation of Forward-Time Finite-Time Lyapunov Exponents Using Discontinuous-Galerkin Spectral Element Methods." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-64997.

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We present an algorithm for computing forward-time finite-time Lyapunov exponents (FTLEs) using discontinuous-Galerkin (DG) operators. Passive fluid tracers are initialized at Gauss-Lobatto quadrature nodes and advected concurrently with direct numerical simulation (DNS) using DG spectral element methods. The flow map is approximated by a high-order polynomial and the deformation gradient tensor is then determined by the spectral derivative. Since DG operators are used to compute the deformation gradient, the algorithm is high-order accurate and is consistent with the DG methods used to compute the fluid solution. The method is validated with a benchmark of a periodic gyre, a vortex advected in uniform flow and the flow around a square cylinder. An exact equation for the FTLE of the advected vortex is derived.

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Yu, Zexing, Fei Du, and Chao Xu. "Time-Domain Spectral Element Simulation of Lamb Wave Time Reversal Method for Detecting a Breathing Crack in a Plate." In ASME 2019 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/imece2019-10495.

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Abstract Lamb wave is considered as an appropriate approach to detect the cracks in structures. This paper combines an efficient time-domain spectral finite element with time reversal method to develop an efficient breathing crack detection method. In this regard, Gauss-Lobatto-Legendre quadrature rules and penalty function method are carried out to construct an effective and accurate approach. Comparing the computation scales and results of this method and traditional finite element method, the validity and superiority of the proposed model is stressed. The reconstructed signals of two scenarios, intact and impaired structures, are captured. It is concluded that, this approach is capable of detecting breathing cracks. In addition, the influences of the relative depth of the notch and incident region are studied. This research may provide the guidance for experiment configuration and the further study.

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Kuang, J. H., and M. H. Hsu. "Eigen Solutions of an Orthotropy Composite Blade Solved by the Differential Quadrature Method." In ASME Turbo Expo 2002: Power for Land, Sea, and Air. ASMEDC, 2002. http://dx.doi.org/10.1115/gt2002-30423.

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The eigenvalue problem of an orthotropy composite blade is formulated by employing the differential quadrature method (DQM). The Euler-Bernoulli beam model is used to characterize the orthotropy composite blade. The differential quadrature method is used to transform the partial differential equations of an orthotropy composite blade into a discrete eigenvalue problem. The Chebyshev-Gauss-Lobatto sample point equation is used to select the sample points. In this study, the effects of the fiber orientation, internal damping, external damping, inclined angle and the rotation speed on the eigen solutions for an orthotropy composite blade are investigated. The effect of the number of sample points on the accuracy of the calculated natural frequencies is also discussed. The integrity and computational efficiency of the DQM in this problem will be demonstrated through a number of case studies. Numerical results indicated that the differential quadrature method is valid and efficient for an orthotropy composite blade formulation.

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Dabiri, Arman, and Eric A. Butcher. "Fractional Chebyshev Collocation Method for Solving Linear Fractional-Order Delay-Differential Equations." In ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/detc2017-68333.

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An efficient numerical method, the fractional Chebyshev collocation method, is proposed for obtaining the solution of a system of linear fractional order delay-differential equations (FDDEs). It is shown that the proposed method overcomes several limitations of current numerical methods for solving linear FDDEs. For instance, the proposed method can be used for linear incommensurate order fractional differential equations and FDDEs, has spectral convergence (unlike finite differences), and does not require a canonical form. To accomplish this, a fractional differentiation matrix is derived at the Chebyshev-Gauss-Lobatto collocation points by using the discrete orthogonal relationship of the Chebyshev polynomials. Then, using two proposed discretization operators for matrix functions results in an explicit form of solution for a system of linear FDDEs with discrete delays. The advantages of using the fractional Chebyshev collocation method are demonstrated in two numerical examples in which the proposed method is compared with the Adams-Bashforth-Moulton method.

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Guessous, Laila, and Yuehong Zheng. "A Pseudo-Spectral Numerical Scheme for the Simulation of Convective Flows." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-33083.

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This paper focuses on the development and validation of a pseudo-spectral numerical scheme, based on a variational formulation, for the solution of the three-dimensional, time-dependent governing equations in wall bounded forced and natural convective flows. One of the novel aspects of this numerical scheme is the use of rescaled Legendre-Lagrangian interpolants to represent the velocity and temperature in the vertical direction. These interpolants were obtained by dividing the Legendre Lagrangian interpolants of same order by the square root of the corresponding weight used for Gauss-Lobatto quadrature. By rescaling the interpolants in such a manner, the mass matrix resulting from the variational formulation becomes the identity matrix, thus simplifying the numerical algorithm. Two specific problems have been investigated as part of the validation process: Steady and unsteady channel flow driven by an external streamwise oscillating pressure gradient and Rayleigh Be´nard convection. In all cases, comparison with exact solutions and published results yield excellent agreement.

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